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Every mathematician has only a few tricks (2020)
- tibbar 11mo agoI think this is true for engineers as well! I enjoy getting to know the "theme" of my favorite coworkers over the years. There was: * The fellow who always looked for the simplest hack possible. Give him the most annoying problem, he'd pause, go Wait a minute! and redefine it to have a very easy solution. He typed very slowly, but it didn't really matter. * The one who truly loved code itself. He would climb mountains to find the most elegant, idiomatic way to express any program. Used the very best practices for testing, libraries, all that. He typed very fast. * The former physicist who spent all his time reading obscure mailing lists on his favorite topics. His level of understanding of problems in his domains of interest was incredible. I could go on and on! It's such a fun taxonomy to collect. All of these friends were marvelous at solving their particular flavor of problem. As for myself, I like to think that my "trick" is to spend a long time poking at the structure of a problem. Eventually the solution I was looking for doesn't matter anymore, but the tools I developed along the way are pretty useful to everyone!
- tibbar 11mo agoHere are a few more. * The (brilliant) infrastructure engineer who described his modus operandi as 'I read stuff on Reddit and then try it out.' This engineer is now worth, as a conservative estimate, in the neighborhood of $50 million. So maybe more of us should be doing that. * Another infrastructure engineer, also very effective, who made a habit of booking an external training session (sometimes a series, weekly) for how to set up and integrate every piece of technology we used. * An engineer (this one is quite famous, and we have only interacted professionally a few times) who simply wrote the best comments in the world. Every method was adorned with a miniature essay exploring the problem to be solved, the tradeoffs, the performance, the bits left undone. I think about those comments quite often. As an addendum, though, I will say that the best engineers overall all shared a trait - they kept trying things until they got something working. That alone will take you pretty far.
- malux85 11mo ago“The most successful people have failed more times than you have tried”
- tibbar 11mo agoYes. I like to think that all of the people above could have solved most of the same problems, albeit in wonderfully different styles, but what really guaranteed success was a commitment to just keep at it. Edit to add: Still, the styles matter a lot! For one thing, they greatly influence which problems each person is interested in. Also, the style you solve problems with colors what your final output looks like, which is perhaps more obvious in engineering than in mathematics.
- psychoslave 11mo agoSuccess is relative. If the goal is to never fail, never try is the best strategy. Also the most sure path to finish in the 1% wealthiest is to start in its network. When the game is set to make 99% of players considered as losers in its own terms, the best strategy to have fun at scale is to not care about the highlighted goal. Keep awareness of how rules actually apply, take shortcuts if it feels safe and preferable, always respect human dignity even when nasty players try to make a dirty move agaisnt you, don't let the lowering bare of hate infect one more player.
- malux85 11mo ago>Success is relative. If the goal is to never fail, never try is the best strategy. This is dangerously not true, if you never try, then you are guaranteed to fail to live up to your potential, which is one of the greatest failures of all.
- psychoslave 11mo agoSorry if it wasn't clear and explicit enough, but this wasn't meant to be an advise to not try anything. Just an extrem example of what it might be logically concluded when starting from unsound grounds. Never fail is not a sound goal. That said living up to our potential is not very much better per se. Or at least it seems far too vague to be aptly considered. What forces are going to define what it is to tap relevantly in our potential?
- philipov 11mo agoFor me, it's tracing code/pipelines to figure out how a result was produced, typically in the context of that result being wrong somehow. Go To Definition is the most useful function in any editor. I'm always surprised by how frequently colleagues don't think to do this and are left helpless.
- tibbar 11mo agoThis reminds me of my further theory that everyone needs one 'heavy' and one 'light' technique. The 'light' technique is something that often works well as a heuristic and can be an effective unit of iteration. The 'heavy' technique is something that you can fall back on in difficult cases, something that can reliably solve hard problems, even if it's slow. Sometimes the heavy technique is: just ask someone else. ;)
- TeMPOraL 11mo ago> Sometimes the heavy technique is: just ask someone else. ;) For a lot of people I know, this is the light technique!
- javawizard 11mo agoYou jest, but that's how my sister gets through life, and it's always fascinated me. She's incredibly intelligent, but more importantly she's a phenomenal social networker. She always has someone to call to ask about any question or solve any problem that comes up in life, and she's great at connecting these people with each other when they have problems of their own - so they all want to help her with whatever she needs, just to gain access to a pool of people they themselves can talk to. What do you do with a skillset like that? I honestly don't know - something in leadership, probably, something where finding the right people and setting them to work is the most important skill of the job.
- TeMPOraL 11mo agoThat wasn't in jest. I worked in a place where this was a norm. Nothing was properly documented, instead everyone would just ask and answer questions on chats; somehow, this actually kept velocity high. Found it really hard to adjust to that. I'm the kind of person that prefers to research things on my own, find hard references and understand context. But there, this was the wrong approach.
- huflungdung 11mo ago[dead]
- NedF 11mo ago[dead]
- qnleigh 11mo agoOne of the answers links https://www.tricki.org/ https://www.tricki.org/, which describes itself as a 'Wiki-style site with a large store of useful mathematical problem-solving techniques.' no longer maintained, but looks neat.
- rramadass 11mo agoThank You for pointing that out; that is indeed nice!
- deleted 11mo ago[deleted]
- Atiscant 11mo agoFor most of my computer science PhD the “trick” was just to get the inductive definition to work, and then how to tweak it for the next paper. Or, get enough structuret we can do an “abstract nonsense” proof[0]. [0]:https://ncatlab.org/nlab/show/category+theory#AbstractNonsense https://ncatlab.org/nlab/show/category+theory#AbstractNonsen...
- eigenman 11mo agoThere is a joke in applied mathematics that we’re like Taco Bell. We all use the same six ingredients, mixing them in different ways. For myself, I’ve found several techniques I use over and over again. Some of this is a “when you’re a hammer, everything looks like a nail.” But fundamentally there are only a handful of ideas. One professor of mine once said the only groundbreaking result in the past few decades was compressive sensing.
- ArcHound 11mo agoI remember annoying one of our professors at the university. Whenever he was discussing a problem and found me falling asleep, he called my name. I woke up, said Chinese remainder theorem and with like 90% success rate it solved the problem handily. Yes, it was an Algebra class. We were still surprised how well it worked.
- Sharlin 11mo agoFrom a comment: > These methods can be combined. First generalize the problem, making it more complicated. Then simplify along a different axis. – Stig Hemmer Very relevant to software design too.
- djmips 10mo agoI feel like our minds work a bit like this.
- Animats 11mo agoFeynman wrote in his autobiography that much of his success came from having different mathematical tricks than most of his peers. So when they were stuck, he could sometimes make progress.
- seanhunter 11mo agoInterestingly, one of his famous tricks of computing complex integrals by parameterizing[1] and then differentiating under the integral is known as “Feynman’s trick” in his honour spite of it having been invented by Euler over 250 years before. [1] https://zackyzz.github.io/feynman.html https://zackyzz.github.io/feynman.html
- tux3 11mo agoA popular naming convention in Mathematics is to name things after the second person to discover them. The first person is Leonhard Euler.
- rramadass 11mo agoLeonhard Euler (https://en.wikipedia.org/wiki/Leonhard_Euler https://en.wikipedia.org/wiki/Leonhard_Euler) is the only reason i believe in "extraterrestrial life" which are "more intelligent" than us :-) List of topics named after Leonhard Euler - https://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler https://en.wikipedia.org/wiki/List_of_topics_named_after_Leo...
- nabla9 11mo agoHe had many tricks in limited area. All his his tricks were limited to classical calculus and nineteenth-century mathematics. He didn't do anything fancy. His style was always the same, he just mastered it really well.
- 19f191ty 11mo agoThat is the actual trick. To master one trick so well that you can spot anywhere its hiding. Unfortunately, this takes a long time, which is why most mathematicians can only do it for a couple of tricks in their lifetime
- seanhunter 11mo agoMy dad (an engineer not a mathematician) would use Newton-Raphson[1] to solve basically any problem that wasn’t very obviously linear. When I was a kid, some of my first programming memories were my dad getting me and my brother to implement Newton-Raphson in basic on an HP85a, getting me to implement Newton-Raphson in rpn on an HP calculator, debugging my dad’s (genuinely revolting) basic program[2] which wouldn’t run but (who would have guessed?) used Newton-Raphson to compute something or other. He basically learned the one numerical root-finder and how to evaluate basic second derivatives and he was set for life on all the problems a career in chemical and process engineering could throw at someone. [1] https://sheffield.ac.uk/media/31988/download?attachment https://sheffield.ac.uk/media/31988/download?attachment [2] He learned to program in FORTRAN and lived by the maxim that a determined FORTRAN programmer can write FORTRAN in any language.
- dataflow 11mo agoRelatedly, I've found Newton-Raphson is a great example of an algorithm where Knuth's "I have only proven it correct, not tried it" rears its head very prominently. The obvious implementations can work flawlessly on toy examples and then fail miserably on real-world examples.
- sigmoid10 11mo agoThere's a reason why numeric analysis is still actively studied by research mathematicians. If we could just throw something as simple as newton's method at any nonlinear problem, we'd only need people to learn this once in school and everyone could solve everything.
- seanhunter 11mo agoYeah for that you need Euler’s method… I mean of course Runge-Kutta. … By that I’m of course referring to rk4. … I mean, you have a point. Joking aside I think having a few basic numerical methods like say Newton, rk4, brent root, monte carlo simulation etc in your general toolbox of techniques you know how to do can make you unreasonably effective in a wide range of situations. Just yesterday I had to solve a problem in a relatively small space so I first used a brute force method to check all the feasible solutions and having got the full list of actual solutions out, figured out the analytical solution. It meant I could be very confident that my analytical solution was correct when I had it.
- _f9cu 11mo agoA mathematician tricks.always. There is no point to untrick the trickster. Misuse of your time
- enaaem 11mo agoI like the trick of adding zero to an equation.
- stared 11mo agoMy lecturer was once unable to solve a problem at the blackboard. After trying for some time, he said he needed a short break to go back to his office to look at his notes. He brought the notes, and there he found a hint he had written for himself: "Use a trick"
- brap 11mo agoFor programmers I would say graphs or at least thinking in graphs is a common one. Some would say SAT is also a handy trick but I’ve personally never used it.
- emil-lp 11mo agoSAT, SMT, ILP, MILP.
- zelphirkalt 11mo agoMy impression from math lectures as a CS student was, that you have to know the tricks, otherwise you just sit there dumbfounded, not knowing how you can make progress. Obviously, there will also be some kind of mathematical understanding of things at play, but a lot of the math stuff is pattern recognition and applying tricks. If you don't know them, it becomes very frustrating. I've not ever seen a way of teaching math though, that focuses on teaching learners these tricks or techniques. They always fell out of things on the side or the teaching person assumed you know it, or was surprised, that the learning people were not able to see it, or able to apply some trick. Or they just didn't care, as long as a small portion of the learners somehow were able to solve it, because then it obviously cannot possibly have been their way of teaching, but other students must have been lazy...
- catchcatchcatch 11mo ago[dead]
- mobeets 11mo agoOne surprise of reading this thread was learning that Terry Tao is an active user of Math Overflow!
- fizx 11mo ago- The hard part of a compiler is the parser. Find an existing parser, then emit using that language's web templates. - Your DB query is better in an inverted index. - Consider your data locality carefully.
- ErroneousBosh 11mo agoI do various things where for example I need to convert a linear movement of a control knob to a non-linear value. Quite often I'll look at it and go "aha, it's exponential subtracted from a constant", and then go "aha, that doesn't quite work, it's a polynomial", and then fiddle with that for twice as long. Eventually, "meh, sod it, I'll just sketch out a graph of what I want on paper, turn it into a lookup table, and LERP for the correct value". It's amazing how quickly you get a sense of "I just need to bend that in a little so that if these are both up full it doesn't go absolutely mental", and plan your tables accordingly. Also ROM is cheap these days.
- thr0waway001 11mo agoIn my world every triangle is a right triangle and I my hammer is the Pythagorean Theorem.